Abstract
This paper is concerned with the relationship between sufficiency, invariance and conditional indipendence. Given a sufficient [sigma]-field, two results are obtained about independence and conditional independence of it and the almost invariant [sigma]-field. Moreover, it includes a rigorous proof of a stronger result than a Stein theorem about sufficiency and invariance in its version for statistics and in the discrete case. Somewhat of a strange condition is used to prove this theorem in its original form; several results are given here to clarify it.
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